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\noindent{\bf Math 3226
\hfill 
{\Large Competing Species}
\hfill 
Laboratory  4F}\bigskip\\
{\em Consider the non-linear differential system
\begin{eqnarray*}
x'&=&x(-x-y+70)\\
y'&=&y(-2x-y+a),
\end{eqnarray*}
where $a$ is a real parameter. Assume that $a$ varies between the values -10 and 170.}\bigskip

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\noindent{\bf 1 } Find all equilibrium points of the system. Their location and number will, of course, depend on $a$.\medskip\\
{\bf 2 } Classify the equilibrium points by linearization.\medskip\\
{\bf 3 } At what values of $a$ does the number of equilibrium points change?\medskip\\
{\bf 4 } At what values of $a$ does the ``type" of an  equilibrium  point change (e.g., from sink to saddle)?\medskip\\
{\bf 5 } The system models a situation, where two animal species compete for limited resources. $x(t)$ and $y(t)$ denote the sizes of the two animal populations at time $t$. 

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Explain, why the differential system above might be a reasonable model for such a situation. What is the meaning of the parameter $a$?\medskip\\
{\bf 6 } Can you make predictions about the long-term fate of the animal populations? How does the parameter $a$  affect their fate?
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